English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular Trigonometry >

sin(θ)<= sqrt(3)cos(θ)

Solution

sin (θ) \leq 3 cos (θ)

Solution

- \frac{2 π}{3} + 2 πn \leq θ \leq \frac{π}{3} + 2 πn

Interval Notation

[- \frac{2 π}{3} + 2 πn, \frac{π}{3} + 2 πn]

Decimal

- 2.09439 \dots + 2 πn \leq θ \leq 1.04719 \dots + 2 πn

Solution steps

sin (θ) \leq 3 cos (θ)

Move

3 cos (θ)

to the left side

sin (θ) \leq 3 cos (θ)

Subtract

3 cos (θ)

from both sides

sin (θ) - 3 cos (θ) \leq 3 cos (θ) - 3 cos (θ)

sin (θ) - 3 cos (θ) \leq 0

sin (θ) - 3 cos (θ) \leq 0

Rewrite using trig identities

Divide both sides by

2

\frac{sin ( θ ) - 3 cos ( θ )}{2} \leq \frac{0}{2}

\frac{sin ( θ ) - 3 cos ( θ )}{2} \leq \frac{0}{2} : \frac{1}{2} sin (θ) - \frac{3}{2} cos (θ) \leq 0

\frac{sin ( θ ) - 3 cos ( θ )}{2} \leq \frac{0}{2}

Expand

\frac{sin ( θ ) - 3 cos ( θ )}{2} : \frac{1}{2} sin (θ) - \frac{3}{2} cos (θ)

\frac{sin ( θ ) - 3 cos ( θ )}{2}

Apply the fraction rule:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{sin ( θ ) - 3 cos ( θ )}{2} = \frac{sin ( θ )}{2} - \frac{3 cos ( θ )}{2}

= \frac{sin ( θ )}{2} - \frac{3 cos ( θ )}{2}

= \frac{1}{2} sin (θ) - \frac{3}{2} cos (θ)

Expand

\frac{0}{2} : 0

\frac{0}{2}

Apply rule

\frac{0}{a} = 0, a \neq = 0

= 0

\frac{1}{2} sin (θ) - \frac{3}{2} cos (θ) \leq 0

\frac{1}{2} sin (θ) - \frac{3}{2} cos (θ) \leq 0

\frac{3}{2} = sin (\frac{π}{3})

\frac{1}{2} sin (θ) - sin (\frac{π}{3}) cos (θ) \leq 0

\frac{1}{2} = cos (\frac{π}{3})

cos (\frac{π}{3}) sin (θ) - sin (\frac{π}{3}) cos (θ) \leq 0

Use the following identity:

- cos (s) sin (t) + cos (t) sin (s) = sin (s - t)

sin (θ - \frac{π}{3}) \leq 0

sin (θ - \frac{π}{3}) \leq 0

For

sin (x) \leq a

, if

- 1 < a < 1

then

- π - arcsin (a) + 2 πn \leq x \leq arcsin (a) + 2 πn

- π - arcsin (0) + 2 πn \leq (θ - \frac{π}{3}) \leq arcsin (0) + 2 πn

a \leq u \leq b

then

a \leq u and u \leq b

- π - arcsin (0) + 2 πn \leq θ - \frac{π}{3} and θ - \frac{π}{3} \leq arcsin (0) + 2 πn

- π - arcsin (0) + 2 πn \leq θ - \frac{π}{3} : θ \geq - \frac{2 π}{3} + 2 πn

- π - arcsin (0) + 2 πn \leq θ - \frac{π}{3}

Switch sides

θ - \frac{π}{3} \geq - π - arcsin (0) + 2 πn

Simplify

- π - arcsin (0) + 2 πn : 2 πn - π

- π - arcsin (0) + 2 πn

Use the following trivial identity:

arcsin (0) = 0

x 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arcsin (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} arcsin (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ}

= - π - 0 + 2 πn

- π - 0 + 2 πn = - π + 2 πn

= 2 πn - π

θ - \frac{π}{3} \geq 2 πn - π

Move

\frac{π}{3}

to the right side

θ - \frac{π}{3} \geq 2 πn - π

Add

\frac{π}{3}

to both sides

θ - \frac{π}{3} + \frac{π}{3} \geq 2 πn - π + \frac{π}{3}

Simplify

θ \geq 2 πn - π + \frac{π}{3}

θ \geq 2 πn - π + \frac{π}{3}

Simplify

- π + \frac{π}{3} : - \frac{2 π}{3}

- π + \frac{π}{3}

Convert element to fraction:

π = \frac{π 3}{3}

= - \frac{π 3}{3} + \frac{π}{3}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{- π 3 + π}{3}

Add similar elements:

- 3 π + π = - 2 π

= \frac{- 2 π}{3}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{2 π}{3}

θ \geq - \frac{2 π}{3} + 2 πn

θ - \frac{π}{3} \leq arcsin (0) + 2 πn : θ \leq 2 πn + \frac{π}{3}

θ - \frac{π}{3} \leq arcsin (0) + 2 πn

Simplify

arcsin (0) + 2 πn : 2 πn

arcsin (0) + 2 πn

Use the following trivial identity:

arcsin (0) = 0

x 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 arcsin (x) 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} arcsin (x) 0^{\circ} 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ}

= 0 + 2 πn

0 + 2 πn = 2 πn

= 2 πn

θ - \frac{π}{3} \leq 2 πn

Move

\frac{π}{3}

to the right side

θ - \frac{π}{3} \leq 2 πn

Add

\frac{π}{3}

to both sides

θ - \frac{π}{3} + \frac{π}{3} \leq 2 πn + \frac{π}{3}

Simplify

θ \leq 2 πn + \frac{π}{3}

θ \leq 2 πn + \frac{π}{3}

Combine the intervals

θ \geq - \frac{2 π}{3} + 2 πn and θ \leq 2 πn + \frac{π}{3}

Merge Overlapping Intervals

- \frac{2 π}{3} + 2 πn \leq θ \leq \frac{π}{3} + 2 πn

Popular Examples

tan(x/2-pi/3)>= 0

tan(t)-tan^2(2)+sec^3(t)>0

(2sin^2(2x)-1/2)<= 1/2

cos(x)+sqrt(3)*sin(x)>= sqrt(2)

cos(x/2)>(sqrt(2))/2